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Tutorial 5. Generating Functions & Sum of Independent Random Variables. Generating Functions. Generating functions are tools for studying distributions of R.V.’s in a different domain. (c.f. Fourier transform of a signal from time to frequency domain)

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Tutorial 5

Tutorial 5

Generating Functions & Sum of Independent Random Variables


Generating functions
Generating Functions

  • Generating functions are tools for studying distributions of R.V.’s in a different domain. (c.f. Fourier transform of a signal from time to frequency domain)

  • Moment Generating Function gX (t)=E[etX]

  • Ordinary Generating Function hX (z)= E[ZX]

  • o.g.f. is also called z-transform which is applied to discrete R.V’s only.


Z transform
z-transform

  • We illustrate the use of g.f.’s by z-transform:

  • Let a non-negative discrete r.v. X with p.m.f.

    {pk, k = 0,1,…}, z is a complex no.

  • The z-transform of {pk} is

    hX(z) = p0 + p1z + p2z2+ ……

    =  pkzk

  • It can be easily seen that

     pkzk = E[zX]


Z transform1
z-transform

  • We can obtain many useful properties of r.v. X from hX(z).

  • First, we can observe that

    • hX(0) = p0 + p10+ p202+ …… = p0

    • hX(1) = p0 + p11+ p212+ …… = 1

  • By differentiate hX(z), we can get the mean and variance of X.


Mean by z transform
Mean by z-transform

  • Put z = 1, we get

  •  hX’(1) is the mean of of X.

  • Similarly,


Variance by z transform
Variance by z-transform

  • E[X2] is called the 2nd moment of X.

  • In general, E[Xk] is called the k-th moment of X. We can get E[Xk] from successive derivatives of hX (z).

  • Since Var(X) = E[X2] - E[X]2, we get


Example bernoulli distr
Example - Bernoulli Distr.

  • Find the mean and variance of a Bernoulli distr. by z-transform.

    P(X=1) = p, P(X=0) = 1-p


Example bernoulli distr1
Example - Bernoulli Distr.

  • E[X] = hX’(1) = p


Finding p j from g t and h z
Finding pj from g(t) and h(z)

  • If we know g(t), then we know h(z), then we can find the pj :


P d f of sum of r v s
p.d.f. of sum of R.V.’s

  • Let X , Y be 2 independent continuous R.V.’s

  • The cumulative distribution function (c.d.f) of X+Y:


P d f of sum of r v s1
p.d.f. of sum of R.V.’s

  • By differentiating the above equation, we obtain the p.d.f. of X+Y:

  • fX+Y(a) is the convolution of fX and fY .


M g f of sum of r v s
m.g.f. of sum of R.V.’s

  • On the other hand, the moment generating function of p.d.f. fX is

  • The m.g.f. of fX+Y is:


M g f of sum of r v s1
m.g.f. of sum of R.V.’s

  • We have obtained an important property:

  • If S = X+Y, where X & Y are independent.

  • In general, if

p.d.f.

m.g.f.


Two armed bandit problem
Two-Armed Bandit Problem

  • You are in a casino and confronted by two slot machines. Each machine pays off either one dollar or nothing. The probability that the first machine pays off a dollar is x and that the second machine pays off a dollar is y. We assume that x and y are random numbers chosen independently from the interval [0,1] and unknown to you. You are permitted to make a series of ten plays, each time choosing one machine or the other.


Two armed bandit problem1
Two-Armed Bandit Problem

  • How should you choose to maximize the number of times that you win?

  • Strategies described in Grinstead and Snell(P.170):

    • Play-the-best (calculate the prob. that each machine will pay off at each stage and choose the machine with the higher prob. )

    • Play-the-winner (choose the same machine when we win and switch machines when we lose)


Coursework 01
Coursework 01

  • Modified two-armed bandit problem:

    both unknown prob. vary in a linear manner over the twenty plays,

    Pr(payoff at kth play for machine i) = ai + kbi

    where ai and bi are constants.

  • Make a series of 20 plays

  • Design a simple strategy to maximize the number of times that you win


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